J. Physiol. (1977), 273, pp. 631-645 With 4 text-figurem Printed in Great Britain
631
THE EFFECT OF 2,3-DIPHOSPHOGLYCERATE ON THE OXYGEN DISSOCIATION CURVE OF HUMAN HAEMOGLOBIN
BY P. J. GOODFORD, F. E. NORRINGTON, R. A. PATERSON AND R. WOOTTON From the Department of Biophysics and Biochemistry, Wellcome Research Laboratories, Langley Court, Beckenham, Kent, BR3 3BS
(Received 21 March 1977) SUMMARY
1. Oxygen dissociation curves for concentrated human haemoglobin solutions (1-6 mmol dm- in haem) have been measured by mixing known quantities of oxy- and deoxyhaemoglobin solutions and measuring the resulting partial pressure of oxygen with an oxygen electrode. 2. Observations in the presence of 2,3-diphosphoglycerate support previous conclusions derived from experiments at low haemoglobin concentrations, the validity of which has been questioned. 3. The two affinity state model of Monod, Wyman & Changeux (1965) does not fully describe the actions of 2,3-diphosphoglycerate and a model in which this allosteric effector not only binds preferentially to the T state but also lowers the oxygen affinity of this state gives an improved fit to the data. INTRODUCTION
Spectrophotometric methods have been used for recent determinations of the oxygen dissociation curve of haemoglobin (Benesch, Benesch & Yu, 1968; Imai, Morimoto, Kotani, Watari, Hirata & Kuroda, 1970; Tyuma, Imai & Shimizu, 1973; Imai & Yonetani, 1975a, b). These workers studied low haemoglobin concentrations in order to obtain accurate absorbance values in the visible region of the optical absorption spectrum. At such low concentrations the normally tetrameric haemoglobin molecule tends to dissociate into dimers, particularly at high oxygen pressure (Thomas & Edelstein, 1972). The interpretation of these results is therefore complicated by the existence of varying amounts of dimers, and a recent analysis (Ackers, Johnson, Mills, Halvorson & Shapiro, 1975) of the effects of dimer formation has cast doubt on the validity of conclusions derived from work carried out at low haemoglobin concentrations.
P. J. GOODFORD AND OTHERS Recently Minton & Imai (1974) have shown that the effects of 2,3diphosphoglycerate on the oxygen dissociation curve of dilute haemoglobin solutions do not conform to the two-state theory of allosteric proteins due to Monod et al. (1965). Minton & Imai (1974) postulated the existence of a third conformational state in order to explain the results of Tyuma et al. (1973). However, the data used in their analysis were obtained for a low haem concentration of only 60 #mol dm3. Under these conditions there would be approximately 18 % dimer formation in the oxy form (Ackers et al. 1975) but virtually none in the deoxy form due to the stabilizing effect of the 2,3-diphosphoglycerate on the deoxy tetramer. Also, at this low haemoglobin concentration the proposed three-state model, though conceptually more satisfactory, still does not give a good fit to the data of Tyuma et al. (1973) and the observations appear to be thermodynamically inconsistent (Shulman, Hopfield & Ogawa, 1975; Szabo & Karplus, 1976). It was therefore of interest to determine the effects of 2,3-diphosphoglycerate on the oxygen dissociation curve of haemoglobin at a higher concentration where there would be minimal dissociation of the tetramer, in order to ascertain whether the two-state model is valid under these conditions. The use of an oxygen electrode combined with a mixing method was first described by Haab, Pilper & Rahn (1960) and provides a quick and accurate method of determining dissociation curves (Seaton & Lloyd, 1974). It also carries the advantage that higher haemoglobin concentrations or even whole blood may be easily used, and protein denaturation and oxidation are reduced compared with gasometric methods due to the absence of a gas/liquid interface. This technique was therefore applied in the present study, and the results are used to compare current theories concerning the effect of 2,3-diphosphoglycerate on the oxygen affinity of
632
haemoglobin. METHODS
Haemoglobin 8olutions. A 'stripped' haemoglobin solution containing less than 0-05 mol total phosphate per mol of haemoglobin was prepared from 3-week-old human blood (Paterson, Eagles, Young & Beddell, 1976) and used for the determination of the oxygen dissociation curves. The solution was 1-6 mmol dm-3 in haem based on the cyanmethaemoglobin method (Diagnostic Reagents Ltd, Thame, U.K.) 0'1 mol dm3 in Hepes (N-2-hydroxyethylpiperazine-N'-2-ethane sulphonic acid) buffer and 0-105 mol dm3 in NaCl. Similar solutions containing 2,3-diphosphoglycerate were prepared by substituting the appropriate amount of NaCl by the sodium salt of 2,3-diphosphoglyceric acid to maintain a constant combined ionic strength of 0h 105 mol dm3. The pH of the solutions was 7*46 + 0-01. Oxygen dissociation curve measurement. Predetermined degrees of oxygen saturation of the hemoglobin were obtained by mixing oxy- and deoxyhaemoglobin solutions in different proportions, and the partial pressures of oxygen (Po2) were measured with an oxygen electrode (Edwards & Martin, 1966). The method is
HAEMOGLOBIN AND 2,3-DIPHOSPHOGLYCERATE
633
similar to that of Seaton & Lloyd (1974) and their comments apply. Curves were determined for stripped hemoglobin and haemoglobin in the presence of 0-1, 0-2, 0-5, 1-0, 2-5 and 5-0 mmol dm-L 2,3-diphosphoglycerate with all solutions and equipment stabilized at 37 0C in a thermostated warm room. The apparatus (Fig. 1) consisted of a 10 cm3 glass syringe, made airtight with silicone grease, with a glass sidearm as shown (A) packed with a narrow-bore siliconerubber tube and a no. 16 Luer needle. The syringe was held in a modified syringe holder from a 10 cms 'Zipette', set-volume, automatic dispenser (B) (Jencons Ltd,
Fig. 1. Experimental apparatus (for description see text). Hemel Hempstead, U.K.). A disposable nylon Luer stopcock (C) was fitted to A. The syringe was attached to the cuvette (75 jum3) of a Radiometer oxygen-electrode housing by a short length of small-bore silicone-rubber tubing pushed into the bore of the syringe exit and clamped with a small artery clip (D). The tubes were arranged to minimize mixing outside the syringe barrel chamber itself. The apparatus was mounted (E) so as to avoid heat transfer from the stirring motor (F) and rheostat to the syringe. The stirrer rotated a 1 cm magnetic flea (G) which tumbled about the inside of the syringe promoting efficient mixing. The spring drew the plunger 24
PHY 273
634
P. J. GOODFORD AND OTHERS
from the syringe barrel. The locknut (H) on the piece of threaded rod was set so that the plunger was held at the 5-0 cm8 mark and the locknut (1) was set so that the plunger could be depressed exactly 0-5 cm3. The oxygen electrode, housing and PHM71Mk2 acid base analyzer with po, module were made by Radiometer, Copenhagen. The syringe was calibrated by making appropriate weighings, before and after filling with water. 30 cm3 hemoglobin solution was needed for the determination of each curve. 10 cms in a 250 cm3 flask was gassed with water-saturated, oxygen-free nitrogen at 37 00 on a slowly rocking shaker for ca. 45 min or until fully deoxygenated. The remaining 20 cm3 was allowed to equilibrate with atmospheric oxygen until a reading of 21-33 kPa oxygen, corrected for barometric pressure, was obtained with the calibrated oxygen electrode. The syringe was gassed with nitrogen and 9 cm3 deoxyhaemoglobin was drawn in through D after turning aside the plunger stop B. The apparatus was tilted in order to eject nitrogen bubbles through A and D. The plunger was then pushed in to the 5-0 cm3 mark and the stop B swung back into place. A 10 cm3 disposable syringe (J) full of oxyhaemoglobin was pushed into the stopcock C which was washed through with oxyhaemoglobin into an empty 2 cm3 syringe (K). The solution was stirred for 2 min and 0-5 cm3 was ejected through D to the oxygen electrode to give the partial pressure reading for deoxyhaemoglobin which was zero on the scale used (0-21-33 kPa). D was clipped, and by manipulating the stopcock C, 0-5 cm3 oxyhaemoglobin was drawn into the chamber. The mixture was stirred until the reading was stable (approx. 2 min), 0-5 cm3 was again ejected through the electrode chamber and the partial pressure read. This cycle was repeated about 20 times until the mixture had a partial pressure of oxygen near to that in the air. Calculation of saturation. If the volume of the mixing chamber is x before and y after the ejection of the 0-5 cm3 sample, then the 0 5 cm3 ejected is x-y. Assuming that the oxygen content of the deoxyhaemoglobin is zero and that the oxyhaemoglobin is fully saturated, the oxygen content of the mixture after the ith addition of x -y cm3 of oxyhaemoglobin, Ci, is given by
C, 1+-+ Y (y =
X
in cm3
-
...
-
(y)
~~~~~~~~~(1) 1
O2Cm3 of solution, where C is the oxygen content of the oxyhaemoglobin.
C = Hk+Sp (2) where H = hemoglobin content in mg/cm3 corrected for methaemoglobin content, k = cm3 oxygen bound per mg hemoglobin (Edwards & Martin, 1966), S = volume of oxygen (cm3) dissolved in 1 cm3 of buffer at 1 kPa corrected for barometric pressure and temperature (Sendroy, Dillon & Van Slyke, 1934), p = p02 of oxyhaemoglobin in kPa. Yi, the fractional saturation of the hemoglobin in solution is thus g (3) C,-Sp, -Hk, where pi = po, observed after the ith addition. I, was calculated for each p, value and plotted against p, to give the experimental oxygen dissociation curves.
HAEMOGLOBIN AND 2,3-DIPHOSPHOGLYCERATE
635
RESULTS
E8timatn of error. The protocol for fitting theoretical models to the 2,3-diphosphoglycerate data was determined from a consideration of the errors found for nine replicate determinations of the oxygen dissociation curve of stripped haemoglobin, and of the source of errors in the experimental technique. The standard errors of partial pressure measurement expressed as a percentage of the measurement are plotted in Fig. 2 6 I '5
2 . 3
. *
* .0
.0
*
CD
5
-
10 15 ~~~~~No.of additions
20
Fig. 2. Ordinate: standard errors of the po,, values expressed as a percentage of the mean po, value at each addition for nine replicate determinations of the oxygen dissociation curve of concentrated stripped haemoglobin solu-
tions at 37 'C. Abscissa: number of additions of oxryhaemoglobin solution to the mixture.
against the number of additions of oxyhaemoglobin. Saturation values are calculated (eqn. (3)) mainly from the volumes of solutions mixed which are assumed to be error-free. Pressure is also involved in the calculation of saturation values but the product, Sp, is never greater than 3-5 %/ of C,, and so the error in saturation is negligible compared with the error in poll Hence virtually all of the experimental error should appear as error in the po, measurements. Fig. 2 shows that the percentage error in po, measurement reman approximately constant for the first fourteen additions of oxyhaemoglobin, and then rises sharply. Further replicate experiments on haemoglobin solution in the presence of 5 0 mmol dm-,3 2,3-diphosphoglycerate 24-2
P. J. GOODFORD AND OTHERS 636 showed a similar pattern and magnitude of errors, indicating that the error is related to the number of additions and not to the absolute value of the po2. This behaviour of the error can be readily explained on the assumption that there is a bias in the experiment arising from a small error in the volume of oxyhaemoglobin added at each cycle. From the design of the apparatus it is likely that this bias would be the same size and in the same direction at each addition, and such a bias of only 2 % would lead to a very similar pattern and magnitude of errors to those observed in the replicate determinations. Non-linear fitting. Theoretical models were fitted to the data obtained for various 2,3-diphosphoglycerate concentrations (Fig. 3) using a nonlinear least squares fitting procedure described by Powell (1970). The function minimized was given by 7
10
/ij obs. pnj calc.\2 -
(4) b j Iz E j=1 i=1\ Where pij obs. and pij calc. are the observed and calculated po0 values for the ith point on the jth curve, and oaj is the standard deviation of the po0 values for the ith point determined from the replicate stripped haemoglobin curves. It was assumed that the values of ao obtained for the stripped curve were also appropriate to the other curves, and the replicate determinations of the 5 mmol dm-3-2,3-diphosphoglycerate curve supported this assumption. Since the experimental error is localized on the Po,
axis it was necessary to calculate residuals parallel to this axis rather than the more usual saturation axis. However, due to the complexity of the models it was not possible to calculate pijcalc. values directly, and so a second iterative process was used within the main iteration in order to find the residuals. The 'two-state' model. According to the two-state theory for allosteric proteins developed by Monod et al. (1965) the saturation of haemoglobin by oxygen is given by Y a(1+ a) 3+Lca1 +cM)3 (5) (1 + cx)4 +L(1 +cc)4
with a = pKR and c = KTIKR. p is the Po2, KR is the oxygen affinity of the high affinity R state of haemoglobin, KT is the oxygen affinity of the low affinity T state and L is the ratio of unliganded T and R states, i.e. L = TO/RO. In the presence of a free solution concentration, d, of an allosteric effector binding at a single site on the T state the parameter L becomes L' where Li L(I+dKD) (6) =
HAEMOGLOBIN AND 2,3-DIPHOSPHOGLYCERATE 637 and KD is the affinity of the allosteric effector for the T state. For high haemoglobin concentrations it is necessary to calculate the free solution concentration of allosteric effector, d, and solving the appropriate mass action equations gives d2.LKD(1 +ca)4+d.{(1 +ca)4+L(1 +ccX)4+LKD(1 +ccX)4 (H-D)} -
D((1 +)4+L(1 +Ca)4} =
0 (7)
where H is the haemoglobin concentration and D is the total concentration of allosteric effector. It is implicit from the equations that observations at low saturation levels are the most critical in distinguishing between the two-and threestate models, and in view of the larger errors found at higher saturation levels only the first ten points of each dissociation curve were used in fitting the data obtained in the presence of 2,3-diphosphoglycerate. It was also assumed that the oxygen affinity of the R state was not affected by the presence of 2,3-diphosphoglycerate. This seems likely from the very weak binding of 2,3-diphosphoglycerate to the R state (Szabo & Karplus, 1976) and there is a large body of evidence in the literature which suggests that this is approximately true (Perutz, 1976). Following the suggestion of Edelstein (1971) KR was fixed at 17x5 kPa-1 which corresponds to the average oxygen affinity of isolated a- and fl-chains at 25 'C determined by Brunori, Noble, Antonini & Wyman (1966). The data of Imai and others (Tyuma et al. 1973; Imai, 1973) suggests a higher value at 25 0C, but recent work by Imai & Yonetani (1975b) indicates that KR falls with increasing temperature. At 35 'C these authors obtained a value of 17*5 kPa-1 for k4, the binding constant (Adair, 1925) for the fourth oxygen molecule of stripped haemoglobin. These findings therefore provide further support for the K11 value 17-5 kPa-1 which we have used at 37 TC, since KR can be approximately equated to k4. Moreover, the precise value of KR is not critical, because wide variations can be accommodated by Monod et al. (1965) type models with corresponding changes in the value of L, as the shape of the function and the goodness of fit are scarcely changed. Indeed, it is not possible to calculate unique values of L and KR unless extremely accurate results at high oxygen saturation levels are available, and some unexpectedly large values of L and KR reported in the literature (Herzfeld & Stanley, 1974) may arise for this reason. The two-state model described by eqns. (5) and (6) was fitted to the 2,3-diphosphoglycerate data using the procedure prescribed and fitting all seven dissociation curves at different effector concentrations simultaneously. The best fit parameter values, their standard errors calculated from the matrix of partial derivatives (Powell, 1970) and the sum of
638 P. J. GOODFORD AND OTHERS squares of residuals for the regression are given in Table 1. Fig. 3 shows the experimental points and the fitted curves calculated from the parameter values in Table 1. Deviations of the experimental points from the fitted curves are largest at low saturation levels, and also, for the higher 2,3-diphosphoglycerate concentrations, the theoretical curves tend to 100
0-01~~~~~~~~ 0~~~~~~
50 °0A j A S ///
631
THE EFFECT OF 2,3-DIPHOSPHOGLYCERATE ON THE OXYGEN DISSOCIATION CURVE OF HUMAN HAEMOGLOBIN
BY P. J. GOODFORD, F. E. NORRINGTON, R. A. PATERSON AND R. WOOTTON From the Department of Biophysics and Biochemistry, Wellcome Research Laboratories, Langley Court, Beckenham, Kent, BR3 3BS
(Received 21 March 1977) SUMMARY
1. Oxygen dissociation curves for concentrated human haemoglobin solutions (1-6 mmol dm- in haem) have been measured by mixing known quantities of oxy- and deoxyhaemoglobin solutions and measuring the resulting partial pressure of oxygen with an oxygen electrode. 2. Observations in the presence of 2,3-diphosphoglycerate support previous conclusions derived from experiments at low haemoglobin concentrations, the validity of which has been questioned. 3. The two affinity state model of Monod, Wyman & Changeux (1965) does not fully describe the actions of 2,3-diphosphoglycerate and a model in which this allosteric effector not only binds preferentially to the T state but also lowers the oxygen affinity of this state gives an improved fit to the data. INTRODUCTION
Spectrophotometric methods have been used for recent determinations of the oxygen dissociation curve of haemoglobin (Benesch, Benesch & Yu, 1968; Imai, Morimoto, Kotani, Watari, Hirata & Kuroda, 1970; Tyuma, Imai & Shimizu, 1973; Imai & Yonetani, 1975a, b). These workers studied low haemoglobin concentrations in order to obtain accurate absorbance values in the visible region of the optical absorption spectrum. At such low concentrations the normally tetrameric haemoglobin molecule tends to dissociate into dimers, particularly at high oxygen pressure (Thomas & Edelstein, 1972). The interpretation of these results is therefore complicated by the existence of varying amounts of dimers, and a recent analysis (Ackers, Johnson, Mills, Halvorson & Shapiro, 1975) of the effects of dimer formation has cast doubt on the validity of conclusions derived from work carried out at low haemoglobin concentrations.
P. J. GOODFORD AND OTHERS Recently Minton & Imai (1974) have shown that the effects of 2,3diphosphoglycerate on the oxygen dissociation curve of dilute haemoglobin solutions do not conform to the two-state theory of allosteric proteins due to Monod et al. (1965). Minton & Imai (1974) postulated the existence of a third conformational state in order to explain the results of Tyuma et al. (1973). However, the data used in their analysis were obtained for a low haem concentration of only 60 #mol dm3. Under these conditions there would be approximately 18 % dimer formation in the oxy form (Ackers et al. 1975) but virtually none in the deoxy form due to the stabilizing effect of the 2,3-diphosphoglycerate on the deoxy tetramer. Also, at this low haemoglobin concentration the proposed three-state model, though conceptually more satisfactory, still does not give a good fit to the data of Tyuma et al. (1973) and the observations appear to be thermodynamically inconsistent (Shulman, Hopfield & Ogawa, 1975; Szabo & Karplus, 1976). It was therefore of interest to determine the effects of 2,3-diphosphoglycerate on the oxygen dissociation curve of haemoglobin at a higher concentration where there would be minimal dissociation of the tetramer, in order to ascertain whether the two-state model is valid under these conditions. The use of an oxygen electrode combined with a mixing method was first described by Haab, Pilper & Rahn (1960) and provides a quick and accurate method of determining dissociation curves (Seaton & Lloyd, 1974). It also carries the advantage that higher haemoglobin concentrations or even whole blood may be easily used, and protein denaturation and oxidation are reduced compared with gasometric methods due to the absence of a gas/liquid interface. This technique was therefore applied in the present study, and the results are used to compare current theories concerning the effect of 2,3-diphosphoglycerate on the oxygen affinity of
632
haemoglobin. METHODS
Haemoglobin 8olutions. A 'stripped' haemoglobin solution containing less than 0-05 mol total phosphate per mol of haemoglobin was prepared from 3-week-old human blood (Paterson, Eagles, Young & Beddell, 1976) and used for the determination of the oxygen dissociation curves. The solution was 1-6 mmol dm-3 in haem based on the cyanmethaemoglobin method (Diagnostic Reagents Ltd, Thame, U.K.) 0'1 mol dm3 in Hepes (N-2-hydroxyethylpiperazine-N'-2-ethane sulphonic acid) buffer and 0-105 mol dm3 in NaCl. Similar solutions containing 2,3-diphosphoglycerate were prepared by substituting the appropriate amount of NaCl by the sodium salt of 2,3-diphosphoglyceric acid to maintain a constant combined ionic strength of 0h 105 mol dm3. The pH of the solutions was 7*46 + 0-01. Oxygen dissociation curve measurement. Predetermined degrees of oxygen saturation of the hemoglobin were obtained by mixing oxy- and deoxyhaemoglobin solutions in different proportions, and the partial pressures of oxygen (Po2) were measured with an oxygen electrode (Edwards & Martin, 1966). The method is
HAEMOGLOBIN AND 2,3-DIPHOSPHOGLYCERATE
633
similar to that of Seaton & Lloyd (1974) and their comments apply. Curves were determined for stripped hemoglobin and haemoglobin in the presence of 0-1, 0-2, 0-5, 1-0, 2-5 and 5-0 mmol dm-L 2,3-diphosphoglycerate with all solutions and equipment stabilized at 37 0C in a thermostated warm room. The apparatus (Fig. 1) consisted of a 10 cm3 glass syringe, made airtight with silicone grease, with a glass sidearm as shown (A) packed with a narrow-bore siliconerubber tube and a no. 16 Luer needle. The syringe was held in a modified syringe holder from a 10 cms 'Zipette', set-volume, automatic dispenser (B) (Jencons Ltd,
Fig. 1. Experimental apparatus (for description see text). Hemel Hempstead, U.K.). A disposable nylon Luer stopcock (C) was fitted to A. The syringe was attached to the cuvette (75 jum3) of a Radiometer oxygen-electrode housing by a short length of small-bore silicone-rubber tubing pushed into the bore of the syringe exit and clamped with a small artery clip (D). The tubes were arranged to minimize mixing outside the syringe barrel chamber itself. The apparatus was mounted (E) so as to avoid heat transfer from the stirring motor (F) and rheostat to the syringe. The stirrer rotated a 1 cm magnetic flea (G) which tumbled about the inside of the syringe promoting efficient mixing. The spring drew the plunger 24
PHY 273
634
P. J. GOODFORD AND OTHERS
from the syringe barrel. The locknut (H) on the piece of threaded rod was set so that the plunger was held at the 5-0 cm8 mark and the locknut (1) was set so that the plunger could be depressed exactly 0-5 cm3. The oxygen electrode, housing and PHM71Mk2 acid base analyzer with po, module were made by Radiometer, Copenhagen. The syringe was calibrated by making appropriate weighings, before and after filling with water. 30 cm3 hemoglobin solution was needed for the determination of each curve. 10 cms in a 250 cm3 flask was gassed with water-saturated, oxygen-free nitrogen at 37 00 on a slowly rocking shaker for ca. 45 min or until fully deoxygenated. The remaining 20 cm3 was allowed to equilibrate with atmospheric oxygen until a reading of 21-33 kPa oxygen, corrected for barometric pressure, was obtained with the calibrated oxygen electrode. The syringe was gassed with nitrogen and 9 cm3 deoxyhaemoglobin was drawn in through D after turning aside the plunger stop B. The apparatus was tilted in order to eject nitrogen bubbles through A and D. The plunger was then pushed in to the 5-0 cm3 mark and the stop B swung back into place. A 10 cm3 disposable syringe (J) full of oxyhaemoglobin was pushed into the stopcock C which was washed through with oxyhaemoglobin into an empty 2 cm3 syringe (K). The solution was stirred for 2 min and 0-5 cm3 was ejected through D to the oxygen electrode to give the partial pressure reading for deoxyhaemoglobin which was zero on the scale used (0-21-33 kPa). D was clipped, and by manipulating the stopcock C, 0-5 cm3 oxyhaemoglobin was drawn into the chamber. The mixture was stirred until the reading was stable (approx. 2 min), 0-5 cm3 was again ejected through the electrode chamber and the partial pressure read. This cycle was repeated about 20 times until the mixture had a partial pressure of oxygen near to that in the air. Calculation of saturation. If the volume of the mixing chamber is x before and y after the ejection of the 0-5 cm3 sample, then the 0 5 cm3 ejected is x-y. Assuming that the oxygen content of the deoxyhaemoglobin is zero and that the oxyhaemoglobin is fully saturated, the oxygen content of the mixture after the ith addition of x -y cm3 of oxyhaemoglobin, Ci, is given by
C, 1+-+ Y (y =
X
in cm3
-
...
-
(y)
~~~~~~~~~(1) 1
O2Cm3 of solution, where C is the oxygen content of the oxyhaemoglobin.
C = Hk+Sp (2) where H = hemoglobin content in mg/cm3 corrected for methaemoglobin content, k = cm3 oxygen bound per mg hemoglobin (Edwards & Martin, 1966), S = volume of oxygen (cm3) dissolved in 1 cm3 of buffer at 1 kPa corrected for barometric pressure and temperature (Sendroy, Dillon & Van Slyke, 1934), p = p02 of oxyhaemoglobin in kPa. Yi, the fractional saturation of the hemoglobin in solution is thus g (3) C,-Sp, -Hk, where pi = po, observed after the ith addition. I, was calculated for each p, value and plotted against p, to give the experimental oxygen dissociation curves.
HAEMOGLOBIN AND 2,3-DIPHOSPHOGLYCERATE
635
RESULTS
E8timatn of error. The protocol for fitting theoretical models to the 2,3-diphosphoglycerate data was determined from a consideration of the errors found for nine replicate determinations of the oxygen dissociation curve of stripped haemoglobin, and of the source of errors in the experimental technique. The standard errors of partial pressure measurement expressed as a percentage of the measurement are plotted in Fig. 2 6 I '5
2 . 3
. *
* .0
.0
*
CD
5
-
10 15 ~~~~~No.of additions
20
Fig. 2. Ordinate: standard errors of the po,, values expressed as a percentage of the mean po, value at each addition for nine replicate determinations of the oxygen dissociation curve of concentrated stripped haemoglobin solu-
tions at 37 'C. Abscissa: number of additions of oxryhaemoglobin solution to the mixture.
against the number of additions of oxyhaemoglobin. Saturation values are calculated (eqn. (3)) mainly from the volumes of solutions mixed which are assumed to be error-free. Pressure is also involved in the calculation of saturation values but the product, Sp, is never greater than 3-5 %/ of C,, and so the error in saturation is negligible compared with the error in poll Hence virtually all of the experimental error should appear as error in the po, measurements. Fig. 2 shows that the percentage error in po, measurement reman approximately constant for the first fourteen additions of oxyhaemoglobin, and then rises sharply. Further replicate experiments on haemoglobin solution in the presence of 5 0 mmol dm-,3 2,3-diphosphoglycerate 24-2
P. J. GOODFORD AND OTHERS 636 showed a similar pattern and magnitude of errors, indicating that the error is related to the number of additions and not to the absolute value of the po2. This behaviour of the error can be readily explained on the assumption that there is a bias in the experiment arising from a small error in the volume of oxyhaemoglobin added at each cycle. From the design of the apparatus it is likely that this bias would be the same size and in the same direction at each addition, and such a bias of only 2 % would lead to a very similar pattern and magnitude of errors to those observed in the replicate determinations. Non-linear fitting. Theoretical models were fitted to the data obtained for various 2,3-diphosphoglycerate concentrations (Fig. 3) using a nonlinear least squares fitting procedure described by Powell (1970). The function minimized was given by 7
10
/ij obs. pnj calc.\2 -
(4) b j Iz E j=1 i=1\ Where pij obs. and pij calc. are the observed and calculated po0 values for the ith point on the jth curve, and oaj is the standard deviation of the po0 values for the ith point determined from the replicate stripped haemoglobin curves. It was assumed that the values of ao obtained for the stripped curve were also appropriate to the other curves, and the replicate determinations of the 5 mmol dm-3-2,3-diphosphoglycerate curve supported this assumption. Since the experimental error is localized on the Po,
axis it was necessary to calculate residuals parallel to this axis rather than the more usual saturation axis. However, due to the complexity of the models it was not possible to calculate pijcalc. values directly, and so a second iterative process was used within the main iteration in order to find the residuals. The 'two-state' model. According to the two-state theory for allosteric proteins developed by Monod et al. (1965) the saturation of haemoglobin by oxygen is given by Y a(1+ a) 3+Lca1 +cM)3 (5) (1 + cx)4 +L(1 +cc)4
with a = pKR and c = KTIKR. p is the Po2, KR is the oxygen affinity of the high affinity R state of haemoglobin, KT is the oxygen affinity of the low affinity T state and L is the ratio of unliganded T and R states, i.e. L = TO/RO. In the presence of a free solution concentration, d, of an allosteric effector binding at a single site on the T state the parameter L becomes L' where Li L(I+dKD) (6) =
HAEMOGLOBIN AND 2,3-DIPHOSPHOGLYCERATE 637 and KD is the affinity of the allosteric effector for the T state. For high haemoglobin concentrations it is necessary to calculate the free solution concentration of allosteric effector, d, and solving the appropriate mass action equations gives d2.LKD(1 +ca)4+d.{(1 +ca)4+L(1 +ccX)4+LKD(1 +ccX)4 (H-D)} -
D((1 +)4+L(1 +Ca)4} =
0 (7)
where H is the haemoglobin concentration and D is the total concentration of allosteric effector. It is implicit from the equations that observations at low saturation levels are the most critical in distinguishing between the two-and threestate models, and in view of the larger errors found at higher saturation levels only the first ten points of each dissociation curve were used in fitting the data obtained in the presence of 2,3-diphosphoglycerate. It was also assumed that the oxygen affinity of the R state was not affected by the presence of 2,3-diphosphoglycerate. This seems likely from the very weak binding of 2,3-diphosphoglycerate to the R state (Szabo & Karplus, 1976) and there is a large body of evidence in the literature which suggests that this is approximately true (Perutz, 1976). Following the suggestion of Edelstein (1971) KR was fixed at 17x5 kPa-1 which corresponds to the average oxygen affinity of isolated a- and fl-chains at 25 'C determined by Brunori, Noble, Antonini & Wyman (1966). The data of Imai and others (Tyuma et al. 1973; Imai, 1973) suggests a higher value at 25 0C, but recent work by Imai & Yonetani (1975b) indicates that KR falls with increasing temperature. At 35 'C these authors obtained a value of 17*5 kPa-1 for k4, the binding constant (Adair, 1925) for the fourth oxygen molecule of stripped haemoglobin. These findings therefore provide further support for the K11 value 17-5 kPa-1 which we have used at 37 TC, since KR can be approximately equated to k4. Moreover, the precise value of KR is not critical, because wide variations can be accommodated by Monod et al. (1965) type models with corresponding changes in the value of L, as the shape of the function and the goodness of fit are scarcely changed. Indeed, it is not possible to calculate unique values of L and KR unless extremely accurate results at high oxygen saturation levels are available, and some unexpectedly large values of L and KR reported in the literature (Herzfeld & Stanley, 1974) may arise for this reason. The two-state model described by eqns. (5) and (6) was fitted to the 2,3-diphosphoglycerate data using the procedure prescribed and fitting all seven dissociation curves at different effector concentrations simultaneously. The best fit parameter values, their standard errors calculated from the matrix of partial derivatives (Powell, 1970) and the sum of
638 P. J. GOODFORD AND OTHERS squares of residuals for the regression are given in Table 1. Fig. 3 shows the experimental points and the fitted curves calculated from the parameter values in Table 1. Deviations of the experimental points from the fitted curves are largest at low saturation levels, and also, for the higher 2,3-diphosphoglycerate concentrations, the theoretical curves tend to 100
0-01~~~~~~~~ 0~~~~~~
50 °0A j A S ///
